In this section, we explore the linewidth of the stationary BEC emission under constant pumping.
The stationary emission spectrum, \(I_{\bf k}(\omega )\) , of the polaritons with the wave vector \(\bf k\)  is determined by the correlation function \(\langle \hat{a}_{\bf k}^\dag (t) \hat{a}_{\bf k}(t+\tau )\rangle \)  [1]}, [2]}
\( I_{\bf k}(\omega )=\lim _{t\rightarrow +\infty }{\rm Re} \int _0^{+\infty } \langle \hat{a}_{\bf k}^\dag (t) \hat{a}_{\bf k}(t+\tau )\rangle e^{-i\omega \tau } d\tau .\) 
